The value of tax shields is not equal to the present value of tax shields: a correction



Abstract

I correct some expressions of Fernandez (2004) and provide a more general expression for the value of tax shields. This expression is the difference between the present values of two different cash flows, each with its own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company. The value of tax shields in a world with no leverage cost is the tax rate times the current debt, plus the tax rate times the present value of the net increases of debt. The value of tax shields depends only on the nature of the stochastic process of the net increase of debt; it does not depend on the nature of the stochastic process of the free cash flow.

I provide a more general expression for the value of tax shields than that in Fernandez (2004). The title of Fernandez (2004) still applies: The value of tax shields is not equal to the present value of tax shields, but the difference between the present values of two different cash flows, each with their own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company. This correction shows that ome of the conclusions of Fernandez (2004) are valid only for specific situations. More specifically, formula (28 VTS = PV[Ku; D·T·Ku] ) is valid only under the assumption that the debt increases are as risky as the free cash flows.

1. Correction of formulae

For simplicity, in Fernandez (2004) I neglected to include in equations (5) to (14) terms with expected value equal to zero. And I wrongly considered as being zero the present value of a variable with expected value equal to zero. This does not have to be the case in general. Because of that error Equations (5) to (17), Tables 3 and 4, and Figure 1 of Fernandez (2004) are correct only if

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Gu is the present value of the taxes paid by the unleve red company and GL is the present value of the taxes paid by the levered company.

The value of tax shield (VTS) comes from the difference between (11a) and (14a):

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4. Conclusions

I show that the value of tax shields in a world with no leverage cost is the tax rate times the debt, plus the tax rate times the present value of the net increases of debt. This expression is the difference between the present values of two different cash flows, each with its own risk: the present value of taxes for the unlevered company and the present value of taxes for the levered company . The critical parameter for calculating the value of tax shields is the present value of the net increases of debt. It may vary for different companies, but in some special circumstances it may be calculated.

For perpetual debt, the value of tax shields is equal to the tax rate times the value of debt. When the company is expected to repay the current debt without issuing new debt, Myers (1974) applies, and the value of tax shields is the present value of the interest times the tax rate, discounted at the required return to debt. If the correct discount rate for the increases of debt is the required return to the unlevered company, then formula (28) of Fernandez (2004) applies.

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REFERENCES

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Fernandez, P., 2004. The value of tax shields is NOT equal to the present value of tax shields.

Journal of Financial Economics 73/1, 145-165.

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Miles, J.A., Ezzell, J.R., 1980. The weighted average cost of capital, perfect capital markets and project life: a clarification. Journal of Financial and Quantitative Analysis 15, 719-730.

Modigliani, F., Miller, M., 1963. Corporate income taxes and the cost of capital: a correction.

American Economic Review 53, 433 -443.

Myers, S.C., 1974. Interactions of corporate financing and investment decisions – implications for capital budgeting. Journal of Finance 29, 1 -25.

Rubac k, R., 1995. A note on capital cash flow valuation. Harvard Business School Case No. 9-

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Management 31, 85–103.

I thank my colleagues José Manuel Campa and Charles Porter for their wonderful help revising earlier manuscripts of this paper, and an anonymous referee for very helpful comments. I also thank Rafael Termes and my colleagues at IESE for their sharp questions that encouraged me to explore valuation problems.

[*]Contact information:

IESE Business School, University of Navarra. Camino del Cerro del Aguila 3. 28023 Madrid, Spain.

E-mail: fernandezpa@iese.edu

[1] I will refer with an “a” to the equations of Fernandez (2004) that are affected by the mentioned error.

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[5] The present value of the principal repayment at the end of year 1 is D /(1+Kd)

The present value of the principal repayment at the end of year 2 is D/[(1+Kd)(1+ K ND)] The present value of the principal repayment at the end of year t is D/[(1+Kd)(1+ KND)t -1] Because D = E{D t}, being Dt the debt repayment at the end of year t.

Previously published by the IESE Business School, May 2005

From PricewaterhouseCoopers Professor of Corporate Finance University of Navarra - IESE Business School, Spain